# Questions tagged [complex-dynamics]

This tag is for questions relating to complex dynamics, study of dynamical systems defined by iteration of functions on complex number spaces. It was an area of research established by Fatou and Julia towards the beginning of the last century.

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### Example of a buried Julia component of a transcendental meromorphic function.

We know examples of buried Julia components (Definition: A Julia component is called buried if it is not contained in the boundary of any Fatou component) for rational functions. In 1998, McMullen ...
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### Equivalent description of Julia set

Let $f$ be a rational map acting on the Riemann sphere $\widehat{\mathbb{C}}$. The Julia set $\mathcal{J}_f$ is the complement of the Fatou set $\mathcal{F}_f$, defined to be the union of all open ...
1 vote
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### Problem about finite grand orbits

I am attempting an exercise in Milnor's Dynamics in One Complex Variable which I have slightly rephrased below: Let $f \in \mathbb{C}(z)$ be a rational function of degree $d \geq 2$. Prove: $0, \infty$...
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### Complex Dynamics book Recommendation

I know the basics of Complex Analysis and Topology and I would like to learn Complex Dynamics. One book I found was Beardon's Iteration of Rational Functions. I'm not sure whether its a good book for ...
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### Mandelbrot set Proof for Bounding Circle of Period 2 Bulb

I've been searching and I can't find a proof for the bounding circle of the Period 2 Bulb in a Mandelbrot set. Its referenced quite a bit that it is a circle with radius of $\frac{1}{4}$ and a centre ...
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### Finite critical points of a polynomial are preperiodic implies the Fatou set is connected and simply connected

I'm currently going through Alan Beardon's book "Iteration of Rational Functions" and I'm a little stuck on his explanation of Corollary 9.5.3. which states that "If every finite ...
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### Conjugating Branch Points $f=ghg^{-1}$

Consider the function $f:=ghg^{-1}$ on $\widetilde{\mathbb{C}}$ where $g$ is a homeomorphism and $h$ is a rational map. Why is it true that branch points of $h$ are transformed by $g$ to removeable ...
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### Rational Mappings of the Annulus

Suppose $R:\widetilde{\mathbb{C}} \rightarrow \widetilde{\mathbb{C}}$ where $R(A) = B$ is a rational mapping from one annulus to another. Assume that one of the components of the complement of $A$ has ...
1 vote
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### Extending a covering map off an annulus

Let $R$ be a map from the Riemann sphere to itself, upon which its restriction to an annulus $A$ is a covering map to another annulus $B$. Suppose there are critical points in one of the complementary ...
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### Finite Number of Periodic Points in the Julia Set

I'm working through Sullivan's proof to his no wandering theorem, and in one of his sections he claims that the set of points of lowest period in the Julia set is finite. I am struggling to see why ...
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1 vote
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### Sign of $n$ th derivative of $f(x)$?

Let $f(z)$ satisfy $f(f(z)) = \operatorname{arcsinh}(z/2)$ More precisely, we construct such an $f(z)$ by using the fixpoint at $0$ and the related Koenigs function. see : https://en.wikipedia.org/...
1 vote
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### Introducing undergraduate students to dynamical systems

In my department a course on dynamical systems is offered this semester. It is a course offered to third (out of four) year undergraduate students and it involves basic dynamics of real maps, ...
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### Equicontinuous Family at a Point

I am reading the definition of equicontinuous family at a point from book called "Iteration of Rational Functions" by Alan F. Beardon. There it is written that equicontinuity of family at a ...
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### Construction of Mandelbrot Set

I am doing Master's project in Complex Dynamics. Here I want to talk particularly about the Mandelbrot set. I have studied its formation and dynamics as parameter $c$ changes (which is very hand ...
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### Examples of Hyperbolic Set and J-Stable sets

I am reading the research article "The Hausdorff dimension of the boundary of Mandelbrot set and Julia sets" by Shishikura. The following two definitions are given without any examples in ...
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### Hyperbolic Set of Extended Complex Plane

I am studying one of the research article "The Hausdorff Dimension of the Boundary of the Mandelbrot Set and Julia Sets" by Mitsuhiro Shishikura. There is one section where he gave the ...
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### Math function for plotting a surface that looks like crumpled paper?

Is there a way to draw a two-dimensional or three-dimensional surface that resembles this? All I know so far is that the dynamics of paper are more complicated than they sound, does this mean I could ...
1 vote
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### Are there any points on the parameter plane that do not belong to any wake?

p/q-wake is the region of parameter plane enclosed by two external rays landing on the same root point on the boundary of Mandelbrot set main cardioid (period 1 hyperbolic component). Are there any ...
1 vote